# Asymmetric relation

In mathematics an **asymmetric relation** is a binary relation on a set *X* where:

- For all
*a*and*b*in*X*, if*a*is related to*b*, then*b*is not related to*a*.^{[1]}

In mathematical notation, this is:

## Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski's axioms characterizing the real numbers **R** is that < over **R** is asymmetric.

An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, every transitive asymmetric relation is a strict partial order.

Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.

The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ x and both are true. In general, any relation in which *x* R *x* holds for some *x* (that is, which is not irreflexive) is also not asymmetric.

Asymmetric is not the same thing as "not symmetric": a relation can be neither symmetric nor asymmetric, such as ≤, or can be both, only in the case of the empty relation (vacuously).

## Properties

- A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
^{[2]} - Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < from the reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.
- A transitive relation is asymmetric if and only if it is irreflexive:
^{[3]}if*a*R*b*and*b*R*a*, transitivity gives*a*R*a*, contradicting irreflexivity.